Why use a singleton instead of static methods?

Often, singletons are used to introduce some kind of global state to an application. (More often than really necessary, to be honest, but that's a topic for another time.)

However, there are a few corner cases where even a stateless singleton can be useful:

• You expect to extend it with state in the foreseeable future.
• You need an object instance for some particular technical reason.
  Example: Synchonization objects for the C# lock or the Java synchronized statement.
• You need inheritance, i.e., you want to be able to easily replace your singleton with another one using the same interface but a different implementation.
  Example: The Toolkit.getDefaultToolkit() method in Java will return a singleton whose exact type is system dependent.
• You want reference equality for a sentinel value.
  Example: DBNull.Value in C#.

I could see a case for a stateless singleton being used instead of a static methods class, namely for Dependency Injection.

If you have a helper class of utility functions that you're using directly, it creates a hidden dependency; you have no control over who can use it, or where. Injecting that same helper class via a stateless singleton instance lets you control where and how it's being used, and replace it / mock it / etc. when you need to.

Making it a singleton instance simply ensures that you're not allocating any more objects of the type than necessary (since you only ever need one).